Choose the correct option(s) with regard to mechanism of the following transformation.

The reaction shown is a classic example of the di-\(\pi\)-methane rearrangement, a photochemical reaction. Upon irradiation with UV light, a system containing a conjugated diene and a methylene bridge undergoes a rearrangement that involves:
Let’s evaluate each option:
\[ \boxed{\text{Correct options: (B) and (C)}} \]
Two positively charged particles \(m_1\) and \(m_2\) have been accelerated across the same potential difference of 200 keV. Given mass of \(m_1 = 1 \,\text{amu}\) and \(m_2 = 4 \,\text{amu}\). The de Broglie wavelength of \(m_1\) will be \(x\) times that of \(m_2\). The value of \(x\) is _______ (nearest integer). 
Structures of four disaccharides are given below. Among the given disaccharides, the non-reducing sugar is: 
The ratio of the fundamental vibrational frequencies \( \left( \nu_{^{13}C^{16}O} / \nu_{^{12}C^{16}O} \right) \) of two diatomic molecules \( ^{13}C^{16}O \) and \( ^{12}C^{16}O \), considering their force constants to be the same, is ___________ (rounded off to two decimal places).
Courage : Bravery :: Yearning :
Select the most appropriate option to complete the analogy.
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
